A Three-Factor Model of Engagement and Persistence From Difficulty-Adjusted Practice Decisions in Online Mathematics Learning

Engagement and persistence drive learning in online environments. Past research often proxied persistence with single, coarse engagement measures, such as time on task. However, recent research indicated more nuanced learner behaviors associated with productive and unproductive struggle. Here, we propose a longitudinal view based on log data in which engagement and persistence are expressed as repeated choices to start, continue, quit, or retry. Using two years of logs from bettermarks, an online environment for learning mathematics (N=2 129 students; >780 000 problem-set records), we estimated student propensities using Rasch-style models that adjust for content difficulty. By applying cross-validated factor analysis on propensities, reflecting decisions to start, continue, quit, and retry while working on problem sets, we identified three factors to predict future learning behavior. Completion propensity predicts retrying and the selection of harder content despite lower accuracy; error-free completion predicts higher first-attempt accuracy alongside a preference for easier content; and activity predicts retry behavior while being associated with lower accuracy. The three-factor model explained substantially more variance than a unidimensional model in predicting future first-attempt accuracy (25% versus 3%, respectively) and student-chosen content easiness (18% versus 6%, respectively), although not retry behavior after failure. We discuss implications for effort-aware adaptivity and dashboards, contributing log-based indicators that can be computed for numerous online learning environments.